The fact that PS, SS, and GSS all use samples from a series of power-posterior
distributions raises some important practical questions:
How many power-posterior distributions are sufficient, how should they be
spaced between the reference and posterior distribution, and how many MCMC
samples are needed from each?
There are no simple answers to these questions, because they will vary depending
on the data and model.
However, one general strategy that is clearly advantageous is having most of
the \(\beta \)  values near zero so that most of the power-posterior distributions
are similar to the reference distribution
[1]}, [2]}, [3]}.
Also, a practical approach to assess if the number of \(\beta \)  values and the
number of samples from each power posterior is sufficient is to estimate the
marginal likelihood multiple times (starting with different seeds for the
random number generator) for each model to get a measure of variance among
estimates.
It is difficult to quantify how much variance is too much, but the estimates
for a model should probably be within a log likelihood unit or two from
each other, and the ranking among models should be consistent.
It can also be useful to check how much the variance among estimates
decreases after repeating the analysis with more \(\beta \)  values
and/or more MCMC sampling from each step;
a large decrease in variance suggests the sampling scheme was insufficient.
